Definition 1.

The domain of expp is restricted because the radius of convergence of the series ∑n=0∞zn/n! over ℂp is precisely r=p-1/(p-1). Recall that, for z∈ℚp, we define

|z|p=1pνp⁢(z)

where νp⁢(z) is the largest exponentν such that pν divides z. For example, if p≥3, then expp is defined over p⁢ℤp. However, e=expp⁡(1) is never defined, but expp⁡(p) is well-defined over ℂp (when p=2, the number e4∈ℂ2 because |4|2=0.25<0.5=r).